Theorem termco 49513

Description: The object of a terminal category. (Contributed by Zhi Wang, 17-Nov-2025.)

Hypotheses

Ref	Expression
termcbas.c	⊢ (𝜑 → 𝐶 ∈ TermCat)
termcbas.b	⊢ 𝐵 = (Base‘𝐶)

Assertion

Ref	Expression
termco	⊢ (𝜑 → ∪ 𝐵 ∈ 𝐵)

Proof of Theorem termco

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		termcbas.c	. . 3 ⊢ (𝜑 → 𝐶 ∈ TermCat)
2		termcbas.b	. . 3 ⊢ 𝐵 = (Base‘𝐶)
3	1, 2	termcbas 49512	. 2 ⊢ (𝜑 → ∃𝑥 𝐵 = {𝑥})
4		unieq 4865	. . . . . 6 ⊢ (𝐵 = {𝑥} → ∪ 𝐵 = ∪ {𝑥})
5		unisnv 4874	. . . . . 6 ⊢ ∪ {𝑥} = 𝑥
6	4, 5	eqtrdi 2782	. . . . 5 ⊢ (𝐵 = {𝑥} → ∪ 𝐵 = 𝑥)
7		vsnid 4611	. . . . 5 ⊢ 𝑥 ∈ {𝑥}
8	6, 7	eqeltrdi 2839	. . . 4 ⊢ (𝐵 = {𝑥} → ∪ 𝐵 ∈ {𝑥})
9		id 22	. . . 4 ⊢ (𝐵 = {𝑥} → 𝐵 = {𝑥})
10	8, 9	eleqtrrd 2834	. . 3 ⊢ (𝐵 = {𝑥} → ∪ 𝐵 ∈ 𝐵)
11	10	exlimiv 1931	. 2 ⊢ (∃𝑥 𝐵 = {𝑥} → ∪ 𝐵 ∈ 𝐵)
12	3, 11	syl 17	1 ⊢ (𝜑 → ∪ 𝐵 ∈ 𝐵)

Syntax hints:   → wi 4   = wceq 1541  ∃wex 1780   ∈ wcel 2111  {csn 4571  ∪ cuni 4854  ‘cfv 6476  Basecbs 17115  TermCatctermc 49504

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-ss 3914  df-nul 4279  df-if 4471  df-sn 4572  df-pr 4574  df-op 4578  df-uni 4855  df-br 5087  df-iota 6432  df-fv 6484  df-termc 49505

This theorem is referenced by: (None)